Antecedentes históricos

The sequential lamination algorithm developed in the foregoing may conveniently be taken as a basis for multiscale simulation in situations in which there is a strict separation of scales: a macroscopic scale characterized by slowly-varying smooth fields; and a much smaller scale commensurate with the size of the evolving microstructure. As remarked by several authors [25, 27, 34, 58], these problems may be solved effectively by pushing the microstructure to thesubgrid scale, while solving the well-posed relaxed problem on the computational grid. In this section we present an example of the application of this multiscale approach in which the macroscopic problem is solved by the finite-element method, while the effective behavior is computed, simultaneously with the macroscopic solution, at the Gauss-point level using the sequential lamination algorithm developed in the foregoing. The particu- lar problem considered concerns the quasistatic normal indentation of a Cu-Al-Ni shape- memory alloy by a spherical indenter. The domain of analysis and the computational mesh are shown in Fig. 2.6. The analysis is reduced to one quarter of the entire domain for sim-

X Y Z

Figure 2.6: Computational domain and finite element mesh.

plicity. In particular, solutions exhibiting broken symmetry are ruled out by the analysis. The size of the computational domain is 20 mm _× 20 mm _× 20 mm. The radius of the indenter is 15 mm. The specimen is fully supported over its entire base, and the remainder of its boundary is free of tractions. The computational mesh contains 254 nodes and 105 ten-node quadratic tetrahedral elements. Contact between the indenter and the specimen is assumed to be frictionless and is enforced by a penalty method [67]. To ensure that the jacobianJ of the deformation remains positive in all variants at all times, the simple energy of each well is augmented by a term of the form [62]

Wvol(J) =        C(J2+J−2₋2)2, J <1 0, otherwise (2.59)

whereCis a constant chosen sufficiently small to minimize the effect on the total energy. By design,Wvol_{(J) and its first and second derivatives vanish at J = 1. In addition, the twin-}

144.7 108.6 72.4 36.2 Energy Well 4 Well 6 Indentation = 0.15 (a) 224.6 168.5 112.3 56.2 Energy Well 4 Well 6 Indentation = 0.375 (b)

Figure 2.7: Cross sections and energy-density contours for two unrelaxed solutions at depths of indentation: a) 0.150 mm, and b) 0.375 mm. The symbols designate the energy well which is activated at each Gauss point of the mesh.

of introducing a lower cutoff for the laminate size and preventing runaway microstructural refinement. In all calculations, the twin-boundary energy per unit area Γ is set to 1 J/m2. The maximum size of the laminate at a particular Gauss point is set to the element size.

The finite-element solution is obtained by dynamic relaxation followed by a precondi- tioned conjugate-gradient iteration [73]. The high level of concurrency in the constitutive calculations was exploited via an MPI-based parallel implementation [66] on the ASCI Blue multiprocessing computer. Performance studies showed excellent load balancing and scala- bility.

Fig. 2.7 shows theunrelaxed deformed configurations, and the corresponding distribution of active energy wells at the Gauss points of the mesh, at depths of indentation of 0.150 and 0.375 mm. As is evident from this figure, two energy wells become active during indentation. The transformed zone under the indenter grows with depth of indentation, but the fineness

Indentation/RIndenter E n e rg y /E 0 0 0.01 0.02 0.03 0 0.001 0.002 0.003 Unrelaxed Laminate Unloading (a) Indentation/RIndenter F o rc e /F 0 0 0.01 0.02 0.03 0 0.1 0.2 0.3 Unrelaxed Laminate Unloading (b)

Figure 2.8: a) Normalized total energy vs. normalized depth of indentation; b) Normalized indentation force vs. normalized depth of indentation.

of the variant arrangement is severely limited by the mesh size. Correspondingly, the total energies and indentation forces recorded during indentation are comparatively high, Fig. 2.8. In this figure the energy has been normalized byE0 =V0C11Austenite, where V0 is the volume

of the undeformed specimen, while the force has been normalized byF0=E0/RIndenter. Therelaxed solution obtained using the sequential lamination algorithm differs markedly from the unrelaxed solution just described, Fig. 2.9. Thus, the relaxed deformation field is accompanied by the development of well-defined microstructures at the subgrid level. Some of the laminates generated by the sequential lamination algorithm are quite complex, reaching rank two. Of note is the appearance of a de-twinned zone directly under the indenter. The effect of relaxation on the total energy and indentation force is quite marked, Fig. 2.8, with the relaxed values lying well below the unrelaxed ones. Unloading exhibits the path-dependent nature of the algorithm, with the microstructure established at maximum

79.7 59.7 39.8 19.9 Energy Rank 0 Rank 1 Rank 2 Indentation = 0.15 (a) 147.7 110.8 73.8 36.9 Energy Rank 0 Rank 1 Rank 2 Indentation = 0.375 (b)

Figure 2.9: Cross section and energy-density contours for relaxed solution at an inden- tation depth of: a) 0.150 mm, and b) 0.375 mm. The symbols indicate the rank of the microstructure at the Gauss points. The insets depict the geometry of the microstructure at the indicated sampling points, with each color representing an individual well, and are of identical size oriented such that the left face corresponds to the cross section plane.

load remaining in place during much of the unloading process, which in turn results in a soft response. The fineness of the microstructure is somewhat overpredicted by the calculations, with some of the variants attaining sub-micron thicknesses. In view of (2.56), this excessive refinement may owe to a low value of the twin-boundary energy Γ, or to an overestimation of the misfit energyWBL_{, or both.}